Giving edges arc shape
So whats going on is because of your model's many edges regular old Bevel won't work. So what you need to do is to go into edge mode and only select the outer most edges, as shown in the first picture. Now when you bevel it shouldn't glitch out with with weird geometry.

Lastly on the second picture I circled in red where two edges  were getting too close together and would cause a problem if beveled a bit more. Essentially watch out for overlapping edges when beveling. 

Hope this helps!




(Also having all those edges is a good thing it means you don't have any N-gons, if you didn't know N-gons are any faces with more then 4 edges. N-gons can cause problems with lighting and texturing and sometimes modeling).